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Documents authored by Fuchs, Janosch

Document
DOI: 10.4230/LIPIcs.STACS.2024.35
The 2-Attractor Problem Is NP-Complete

Authors: Janosch Fuchs and Philip Whittington

Published in: LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

Abstract
A k-attractor is a combinatorial object unifying dictionary-based compression. It allows to compare the repetitiveness measures of different dictionary compressors such as Lempel-Ziv 77, the Burrows-Wheeler transform, straight line programs and macro schemes. For a string T ∈ Σⁿ, the k-attractor is defined as a set of positions Γ ⊆ [1,n], such that every distinct substring of length at most k is covered by at least one of the selected positions. Thus, if a substring occurs multiple times in T, one position suffices to cover it. A 1-attractor is easily computed in linear time, while Kempa and Prezza [STOC 2018] have shown that for k ≥ 3, it is NP-complete to compute the smallest k-attractor by a reduction from k-set cover. The main result of this paper answers the open question for the complexity of the 2-attractor problem, showing that the problem remains NP-complete. Kempa and Prezza’s proof for k ≥ 3 also reduces the 2-attractor problem to the 2-set cover problem, which is equivalent to edge cover, but that does not fully capture the complexity of the 2-attractor problem. For this reason, we extend edge cover by a color function on the edges, yielding the colorful edge cover problem. Any edge cover must then satisfy the additional constraint that each color is represented. This extension raises the complexity such that colorful edge cover becomes NP-complete while also more precisely modeling the 2-attractor problem. We obtain a reduction showing k-attractor to be NP-complete and APX-hard for any k ≥ 2.
Cite as

Janosch Fuchs and Philip Whittington. The 2-Attractor Problem Is NP-Complete. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fuchs_et_al:LIPIcs.STACS.2024.35,
  author =	{Fuchs, Janosch and Whittington, Philip},
  title =	{{The 2-Attractor Problem Is NP-Complete}},
  booktitle =	{41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)},
  pages =	{35:1--35:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-311-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{289},
  editor =	{Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.35},
  URN =		{urn:nbn:de:0030-drops-197457},
  doi =		{10.4230/LIPIcs.STACS.2024.35},
  annote =	{Keywords: String attractors, dictionary compression, computational complexity}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2019.10
The Complexity of Packing Edge-Disjoint Paths

Authors: Jan Dreier, Janosch Fuchs, Tim A. Hartmann, Philipp Kuinke, Peter Rossmanith, Bjoern Tauer, and Hung-Lung Wang

Published in: LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

Abstract
We introduce and study the complexity of Path Packing. Given a graph G and a list of paths, the task is to embed the paths edge-disjoint in G. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard. Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case Exact Path Packing where the paths have to cover every edge in G exactly once is already NP-hard for two paths on 4-regular graphs.
Cite as

Jan Dreier, Janosch Fuchs, Tim A. Hartmann, Philipp Kuinke, Peter Rossmanith, Bjoern Tauer, and Hung-Lung Wang. The Complexity of Packing Edge-Disjoint Paths. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dreier_et_al:LIPIcs.IPEC.2019.10,
  author =	{Dreier, Jan and Fuchs, Janosch and Hartmann, Tim A. and Kuinke, Philipp and Rossmanith, Peter and Tauer, Bjoern and Wang, Hung-Lung},
  title =	{{The Complexity of Packing Edge-Disjoint Paths}},
  booktitle =	{14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
  pages =	{10:1--10:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-129-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{148},
  editor =	{Jansen, Bart M. P. and Telle, Jan Arne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.10},
  URN =		{urn:nbn:de:0030-drops-114710},
  doi =		{10.4230/LIPIcs.IPEC.2019.10},
  annote =	{Keywords: parameterized complexity, embedding, packing, covering, Hamiltonian path, unary binpacking, path-perfect graphs}
}
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